Equation (REF ) can be used to define an “extended likelihood” for
estimation using computational methods for latent variables, where
the observations on \(\mathbf {U}\)  are the latents. This has two
advantages. First, the \(O(2^m)\)  computation at (REF ) is avoided, allowing
estimation for higher values of \(m\) . Second,
only the copula density \(c\)  is required and not the copula function \(C\) , which is
an advantage for some copulas where only \(c\)  can be computed,
as is the case with most vine copulas [1]}, [2]}.
Bayesian data augmentation can be used based on (REF ), and
evaluated using Markov chain Monte Carlo (MCMC)
as in [3]} or variational Bayes methods as in [4]}.
The latter is particularly attractive, because it allows for the estimation of
discrete-margined copulas of very high dimensions, with examples up to \(m=792\)  presented
by these authors.
